2007

Determination of the complete set of statically balanced planar four-bar mechanisms

B. Moore, J. Schicho, C. Gosselin

@techreport{RISC3123, author = {B. Moore and J. Schicho and C. Gosselin}, title = {{Determination of the complete set of statically balanced planar four-bar mechanisms}}, language = {english}, abstract = {In this paper, we present a new method to determine the complete set of statically balanced planar four-bar mechanisms. We formulate the kinematic constraints and the static balancing constraints as algebraic equations over real and complex variables. This leads to the problem of factorization of Laurent polynomials which can be solved using Newton polytopes and Minkowski sums. The result of this process is a set of necessary and sufficient conditions for statically balanced four-bar mechanisms.}, number = {2007-14}, year = {2007}, month = {July}, institution = {SFB F013}, length = {16}}

2006

A Lie algebra method for rational parametrization of Severi–Brauer surfaces

2005

Local Parametrization of Cubic Surfaces

I. Szil\'agyi and B. J\"uttler and J. Schicho

@article{RISC2049, author = {I. Szil\'agyi and B. J\"uttler and J. Schicho}, title = {{Local Parametrization of Cubic Surfaces}}, language = {english}, abstract = {Algebraic surfaces -- which are frequently used in geometricmodelling -- are represented either in implicit or parametricform. Several techniques for parameterizing a rational algebraicsurface as a whole exist. However, in many applications, itsuffices to parameterize a small portion of the surface. Thismotivates the analysis of local parametrizations, i.e.,parametrizations of a small neighborhood of a given point $P$ ofthe surface $S$. In this paper we introduce several techniques forgenerating such parameterizations for nonsingular cubic surfaces.For this class of surfaces, it is shown that the localparametrization problem can be solved for all points, and any suchsurface can be covered completely.}, journal = {Journal of Symbolic Computation}, pages = {1--24}, isbn_issn = {ISSN 0747-7171}, year = {2005}, note = {to appear}, refereed = {yes}, length = {22}}

Implicitization and Distance Bounds

M. Aigner, I. Szil\'agyi, B. J\"uttler, J. Schicho

@inproceedings{RISC2439, author = {M. Aigner and I. Szil\'agyi and B. J\"uttler and J. Schicho}, title = {{Implicitization and Distance Bounds}}, booktitle = {{Mathematic and Visualization}}, language = {english}, abstract = {In this paper, we combine results concerning the numerical stabilityof the implicitization process for a given planar rational curve, withresults on the the stability of the resulting implicit representation.More precisely, it is shown that for any approximate parameterizationof the given curve, the curve obtained by an approximateimplicitization with a given precision is contained within a certainperturbation region. The results can be generalized to the case ofsurfaces.}, series = {Mathematics and Visualization}, pages = {1--14}, publisher = {Springer}, isbn_issn = {?}, year = {2005}, note = {to appear}, editor = {M. Elkadi}, refereed = {yes}, length = {14}}

Symbolic-Numeric Techniques for Cubic Surfaces

Ibolya Szilagyi

@phdthesis{RISC2472, author = {Ibolya Szilagyi}, title = {{Symbolic-Numeric Techniques for Cubic Surfaces}}, language = {english}, abstract = {In geometric modelling and related areas algebraic curves/surfacestypically are described either as the zero set of an algebraicequation (implicit representation), or as the image of amap given by rational functions (parametricrepresentation). The availability of both representations oftenresult in more efficient computations.Computational theories and techniques of algebraic geometry infloating point environment are of high interest in geometricmodelling related communities. Therefore, deriving approximatealgorithms that can be applied to numeric data have become a veryactive research area. In this thesis we focus on the twoconversion problems, called implicitization and parametrization,from the numeric point of view.A very important issue in the implicitization problem is theperturbation behavior of parametric objects. For a numericallygiven parametrization we cannot compute an exact implicitequation, just an approximate one. We introduce a conditionnumber of the implicitization problem to measure the worst effecton the solution, when the input data is perturbed by a smallamount. Using this condition number we study the algebraic andgeometric robustness of the implicitization process.Several techniques for parameterizing a rational algebraic surface as a whole exist. However, in many applications, itsuffices to parameterize a small portion of the surface. Thismotivates the analysis of local parametrizations, i.e.parametrizations of a small neighborhood of a given point $P$ ofthe surface $S$. We introduce several techniques for generatingsuch parameterizations for nonsingular cubic surfaces. For thisclass of surfaces, it is shown that the local parametrizationproblem can be solved for all points, and any such surface can becovered completely.}, year = {2005}, month = {July}, note = {PhD Thesis}, translation = {0}, school = {RISC-Linz}, keywords = {implicitization, numerical stability, local parametrization, cubic surface}, sponsor = {RISC PhD scholarship program of the government of Upper Austria, and by the Spezialforschungsbereich (SFB) grant F1303, Austrian Science Foundation (FWF).}, length = {112}}

Local Parametrization of Cubic Surfaces

I. Szil\'agyi and B. J\"uttler and J. Schicho

@techreport{RISC2047, author = {I. Szil\'agyi and B. J\"uttler and J. Schicho}, title = {{Local Parametrization of Cubic Surfaces}}, language = {english}, abstract = {Algebraic surfaces -- which are frequently used in geometricmodelling -- are represented either in implicit or parametricform. Several techniques for parameterizing a rational algebraicsurface as a whole exist. However, in many applications, itsuffices to parameterize a small portion of the surface. Thismotivates the analysis of local parametrizations, i.e.,parametrizations of a small neighborhood of a given point $P$ ofthe surface $S$. In this paper we introduce several techniques forgenerating such parameterizations for nonsingular cubic surfaces.For this class of surfaces, it is shown that the localparametrization problem can be solved for all points, and any suchsurface can be covered completely.}, number = {2004-31}, year = {2004}, institution = {J.~Kepler University, Linz}, keywords = {parametrization, cubics, algorithm, surfaces}, length = {22}, url = {http://www.sfb013.uni-linz.ac.at/}, type = {SFB-Report}}